Chapter 4 Sect. 1,2,3
Download
Report
Transcript Chapter 4 Sect. 1,2,3
Chapter 4
Techniques of
Differentiation
Sections 4.1, 4.2, and 4.3
Techniques of Differentiation
The Product and Quotient Rules
The Chain Rule
Derivatives of Logarithmic and Exponential
asFunctions
Available Rules for Derivatives
1)
d
c 0
dx
2)
d n
x nx n1
dx
3)
d
d
cf ( x) c f ( x)
dx
dx
4)
d
f x g x f ( x) g ( x)
dx
c a constant
or
or
x
n
c 0
nx n 1
or
cf ( x) cf ( x)
Two More Rules
If f (x) and g (x) are differentiable functions, then
we have
The product rule
5)
d
f x g x f ( x ) g ( x ) f ( x ) g ( x )
dx
The quotient rule
6)
d f x f ( x ) g ( x ) f ( x ) g ( x )
2
dx g ( x )
g ( x )
The Product Rule - Example
If f ( x ) x3 2 x 5 3x7 8 x 2 1 , find f ( x )
2
7
2
3
6
f ( x) 3x 2 3x 8 x 1 x 2 x 5 21x 16 x
Derivative
of first
Derivative
of Second
30 x 48x 105x 40 x 45x 80x 2
9
7
6
4
2
The Quotient Rule - Example
3x 5
If f ( x ) 2
, find f ( x )
x 2
Derivative of
numerator
f ( x)
3 x 2 2 2 x 3x 5
x
2
2
3x2 10 x 6
x
2
2
2
2
Derivative of
denominator
Calculation Thought Experiment
Given an expression, consider the steps you
would use in computing its value. If the last
operation is multiplication, treat the expression
as a product; if the last operation is division,
treat the expression as a quotient; and so on.
Calculation Thought Experiment
Example:
2x 43x 6
To compute a value, first you would evaluate the
parentheses then multiply the results, so this can
be treated as a product.
Example: 2x 43x 6 5x
To compute a value, the last operation would be
to subtract, so this can be treated as a difference.
The Chain Rule
If f is a differentiable function of u and u is a
differentiable function of x, then the composite
f (u) is a differentiable function of x, and
d
du
f (u ) f (u )
dx
dx
The derivative of a f (quantity) is the derivative of f
evaluated at the quantity, times the derivative of the
quantity.
Generalized Power Rule
7)
Example:
d n
n 1 du
u nu
dx
dx
12
d
d
2
2
3x 4 x
3x 4 x
dx
dx
1 2
1
2
3x 4 x
6x 4
2
3x 2
3x 2 4 x
Generalized Power Rule
7
Example: If G( x) 2 x 1 find G( x)
3x 5
6
2 x 1 3x 5 2 2 x 1 3
G( x) 7
2
3
x
5
3x 5
2x 1
7
3x 5
6
13
3x 5
2
91 2 x 1
3x 5
8
6
Chain Rule in Differential
Notation
If y is a differentiable function of u and u is a
differentiable function of x, then
dy dy du
dx du dx
Chain Rule Example
dy
If y u and u 7 x 3 x , find y
dx
dy dy du 5 3 2
u 56 x 7 6 x
dx du dx 2
52
8
2
Sub in for u
15x 7 x 3x
5
7 x8 3 x 2
2
140 x
7
32
56 x 7 6 x
8
2
32
Logarithmic Functions
Derivative of the Natural Logarithm
d
1
ln x
dx
x
x 0
Generalized Rule for Natural Logarithm Functions
If u is a differentiable function, then
d
1 du
ln u
dx
u dx
Examples
Find the derivative of f ( x) ln 2 x 1 .
d 2
2 x 1
4x
dx
2
f ( x)
2
2x 1
2x 1
Find an equation of the tangent line to the graph of
2
f ( x) 2x ln x at 1,2.
Slope:
1
f ( x) 2
x
f (1) 3
Equation:
y 2 3( x 1)
y 3x 1
More Logarithmic Functions
Derivative of a Logarithmic Function.
d
1
log b x
dx
x ln b
Generalized Rule for Logarithm Functions.
If u is a differentiable function, then
d
1 du
logb u
dx
u ln b dx
Examples
d
log 4 x 2 3 4 x
dx
d
log 4 x 2 log 4 3 4 x
dx
1
1
(4)
( x 2) ln 4 (3 4 x) ln 4
Logarithms of Absolute Values
d
1 du
ln u
dx
u dx
d
1 du
logb u
dx
u ln b dx
Examples
d
1
2
ln 8 x 3 2
16 x
dx
8x 3
d
1
1
1
log3 2
2
dx
x
1/ x 2 ln 3 x
1
x 2 x 2 ln 3
Exponential Functions
Derivative of the natural exponential function.
d x
e e x
dx
Generalized Rule for the natural exponential function.
If u is a differentiable function, then
d u
du
u
e e
dx
dx
Examples
35 x
f
(
x
)
e
.
Find the derivative of
f ( x) e
3 5 x
d
35 x
3 5 x 5e
dx
Find the derivative of f ( x) x e
4 4x
f ( x) x e
4 4x
4 4x
4
4x
x e x e
4 x e 4 x e 4 x3e4 x 1 x
3 4x
4 4x
Exponential Functions
Derivative of general exponential functions.
d x
b b x ln b
dx
Generalized Rule for general exponential functions.
If u is a differentiable function, then
d u
du
u
b b ln b
dx
dx
Exponential Functions
Find the derivative of f ( x) 7
f ( x) 7
x2 2 x
x2 2 x
d 2
ln 7
x 2x
dx
2x 2 7
x2 2 x
ln 7